I know that the formula for area between curves is the anti derivative of f(x)-f(g), however i'm unsure of how to take the anti-derivative of the absolute value, please help!

determine the area between the curves |x| and -2x^2 + 5

You should know that |x| = x for x > 0 and |x| = -x for x < 0. From the graph that you drew (and I know you've drawn a graph because your teacher would have taught you that drawing a graph is the very, very first thing that you're meant to do) it should be clear that, by symmetry, the required area is:

where is the positive x-coordinate of the intersection point of and (one of your jobs is to get the value of ).